Finding geodesics in a triangulated 2-sphere

TL;DR

This paper adapts thin position techniques from knot theory to find at least three stable geodesics on a triangulated 2-sphere, improving understanding of geodesic stability in discrete geometry.

Contribution

It introduces a novel application of knot theory methods to identify stable geodesics on triangulated spheres, extending classical results.

Findings

At least three stable geodesics can be found on a triangulated 2-sphere.

The method applies a piece-wise linear approach to improve geodesic stability detection.

The approach generalizes the three geodesics theorem to include stability considerations.

Abstract

Let S be a triangulated 2-sphere with fixed triangulation T. We apply the methods of thin position from knot theory to obtain a simple version of the three geodesics theorem for the 2-sphere [5]. In general these three geodesics may be unstable, corresponding, for example, to the three equators of an ellipsoid. Using a piece-wise linear approach, we show that we can usually find at least three stable geodesics.